Abstract This paper is concerned with the following open problem. Is every directly finite, regular right self-injective ring necessarily left self-injective [2, Open problem 14]? This problem is 21 years old and came from J.-E. Roos . We solve in the negative by giving an example. In the example, we construct a simple regular subring of the rank metric completion of a direct limit of a sequence of matrix rings over a field. Moreover the maximal right quotient ring of this simple regular ring is directly finite and not left self-injective. This result was announced in  without proof.